We consider networks of finite-state machines having local transitions conditioned by the current state of other automata. In this paper, we depict a reduction procedure tailored for a given reachability property of the form ``from global state s there exists a sequence of transitions leading to a state where an automaton g is in a local state T'. By exploiting a causality analysis of the transitions within the individual automata, the proposed reduction removes local transitions while preserving all the minimal traces that satisfy the reachability property. The complexity of the procedure is polynomial in the total number of local states and transitions, and exponential in the number of local states within one automaton. Applied to automata networks modelling dynamics of biological systems, we observe that the reduction shrinks down significantly the reachable state space, enhancing the tractability of the model-checking of large networks.